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So we spent our last lecture in defining the general form of the derivative.

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So now we are able to numerically calculate the derivative of any function, not only the first derivative

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but the derivative.

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So now we can go ahead and define the Taylor expansion so we can right now def Taylor.

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And this time it will really be a general Taylor from the river.

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Sorry, a Taylor expansion of any function and up to and or there.

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And yeah, really the most general thing you can do for Taylor expansions.

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And so now we need quite a lot of arguments.

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So of course we need f, we need X, we need X0 Then we need and max the index word a series will terminate.

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And then also we need the value h because we have the derivatives here.

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So F X and zero are the same as we had before.

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So like, for example, here.

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So I will copy this.

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These are just the comments for the arguments and f will be the function, of course, and age will

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be what we have written here.

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The step size.

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And now we can go ahead and we can basically just program the very first equation in our notebook,

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so let me briefly scroll up here.

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So this equation we have to program and for this derivative here, we have just defined our general

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function.

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So we will of course, call this function now.

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So I will scroll down back to the Taylor code and once again, we have a sum, so I will once again

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do it in a loop, even though this is not really the most elegant way of doing it.

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But I think it's very instructive and you understand much better what is going on.

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And now we, of course, as previously add up the values and then we will return the value of teeth.

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So now we have to program the terms.

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And if you don't remember, you can scroll up in your notebook once again.

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But I remember we have to calculate the derivative.

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So now you just have to be careful to use the correct arguments here.

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So we have function acts, age and so function will, of course, be the same as here then.

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We have zero than we have each, and we have.

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And so here you have to be careful that you don't use and max because and Max is the index in which

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the series will terminate.

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But here we have end, which is the index over which we add.

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So I will just briefly scroll up one more time.

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So here you see the indexes and really so yeah, like this should be correct.

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And then we calculate the difference here x minus x zero, which is the position at which we calculate

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the derivative and we calculate this to the power of RN.

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And then we are only missing the factorial, which is given like this.

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So factorial of and and now I have a problem because I forgot this one.

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And now it could work.

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Let's check.

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At least we don't get an error message.

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So I said already, I want to plot something here, so I want to make a plot pretty similar to this

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one.

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So I would copy this so that we don't have to type so much and don't lose too much time.

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So for the next list, I will go to the limit minus five to five.

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Then for the scatterplot, I will use X list and here I will use our function func than this one.

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Looks good.

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But now, of course, we have to update this one.

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And here we have to just write in general, Taylor and.

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For this one, I will come back later, and for Taylor, we have to, of course, use now the correct

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arguments.

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So we have to right first argument is to function funk, then x list for the X values, and this will

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really be a whole list, not just individual numbers.

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Then for zero, we can really try different values.

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So for example, zero now we can use and Macs and Macs.

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I would say we should take some number like let's start with five and four h.

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We should also test a few values, but for the beginning, let's use zero point one.

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And this looks good.

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But now, just to add a few more of these plots at different x zero positions, I will use two and minus

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three things is similar to what we had previously and change the color red and green.

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And so now we have here our plots.

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Maybe the right y range doesn't look so good.

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That's changing a bit.

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OK, now it looks better.

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So you see these light blue dots here to scatter plots.

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This is the actual function that we have defined.

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This was something like sine square.

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Or is it here two times sine squared plus x and the other functions, the plots here it has lines.

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These are our Taylor expansions at different positions, at the position zero, two and three or minus

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three.

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So you see that at their respective points, for example, here near zero for blue, it works, OK?

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But the range in which it works is not too large.

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So what we would have to do here is probably to increase the number of terms to be taken to account

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for this series.

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So if I increased this number?

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It looks much better.

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So we have a much larger range of X values where the function would series agrees with the function.

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However, the agreement is not really that good.

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So what we can do is we can change the value of h for the derivative.

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And then you see, I would I would say the fit now looks better in this in this very narrow area.

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But this time, of course, again, the range where the fit is OK has decreased.

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So this means decreasing h gives you a better quality for the derivatives, which gives you a better

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accuracy of the series near the expansion point, but it loses you some accuracy far away from the expansion

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point.

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So that's a bit of a trade off here.

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So, of course, you could also increase the number of end max, but then you get sometimes some artifacts,

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so you really have to play them with the values of H and and Max to get really an ideal result here.

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So it can be sometimes a bit difficult.

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All right.

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So but I think overall we have seen that it works quite well to approximate these functions in some

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narrow range around the position X0.

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And we have learned that we can describe it basically any function as a polynomial by just using our

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our expansion formula from the very beginning, F of X is equal to the series of these derivatives times,

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these polynomial terms.

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What I think is also really nice is that we have written all of this in a very general way, so we cannot

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only calculate this Taylor expansion for this particular function, but we can just copy the whole block

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of code here and also copy this line of code where we have defined the function and we can just redefine

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the function here in our new self.

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We can just take another function.

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For example, we could take two times sine squared times and p dot exponential function.

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And then we just use some gauss curve.

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So this time, maybe we should change the range of it.

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This time we have such as sine function or sine squared, but to to say and modify modulated by the

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gauss function.

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And you see in the vicinity of the respective expansion point, the Taylor expansion works quite well.

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And I think this is really a nice tool that we have established here with this definition of the Taylor

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expansion where we have used also this definition of the derivative.